Concept

Cramér's model

A random imitation of the primes in which each number n is declared prime with chance one over its logarithm, independently. It predicts the primes' density and the size of their largest gaps well, but ignores divisibility by small primes.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Ulam's spiral to 900. The integers up to 900 laid out in a square spiral, with the primes marked; they crowd onto diagonal lines.

The primes on a spiral, and a pattern nobody ordered

Wind the whole numbers outward in a square spiral, mark the primes, and they line up on diagonals. The observation is a hundred years old and there is still no proof it means anything.

discrete · Prime distribution

Record gaps between primes against the square of the logarithm. 24 record prime gaps below thirty million as a staircase of points against log p, under the curve (log p)² and a dashed curve 1.12 (log p)²; the last record is 210 after 20831323.

The longest wait for a prime

Near a number x the primes are on average log x apart, and the longest stretch without one below x is much longer: about the square of log x, if the primes behave like random numbers with the right density. Sieving to thirty million finds twenty-four record gaps, all climbing in the shape of that square and all below it. A random model of the primes gets the scale right and the details wrong, and the correction for what it gets wrong predicts gaps larger still — larger than any anyone has found.

discrete · Prime distribution

Named alongside it

The objects these essays reach for when they reach for this one.

Conjecture Density Parity Prime Prime gaps Prime number theorem Primes Quadratic polynomials Random model Sieve Ulam spiral

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